Problem 2a: Solve the system by either the substitution or
elimination by addition method.

2a.

Answer:

I'm going to chose to use the elimination by addition method, however
it would be perfectly fine for you to use the substitution method.
Either way the answer will be the same.

Eliminating z from equations (1) and
(2) we get:

Eliminating z from equations (1) and
(3) we get:

Putting equations (4) and (5) together into a system of two equations
and two unknowns we get:

Eliminating y from equations (4) and
(5) we get:

Solving for x we get:

Plugging 2 in for x in equation (4) and
solving for y we get:

Plugging in 2 for x and -1 for y in equation (1) and solving for z we get:

(2, -1, 3) is the solution to our system.

Problems 3a - 3b: Solve the given word problems using systems
of equations.

3a. It took a boat 4 hours to travel downstream 120 miles. Upstream,
the same trip took 6 hours. Find the rate of the boat in still water
and the rate of the current.

Answer:

Let x = the rate of the boat in still water

Let y = the rate of the current.

(Rate)

(Time)

= Distance

With current

x + y

4

120

Against current

x - y

6

120

Using the fact that (Rate)(Time) = Distance, we get the system:

Simplifying this system we get:

Eliminating the x's we get:

Solving for y we get:

Plugging 5 in for y in the first simplified
equation and solving for x we get:

The speed of the boat in still water is 25 mph and the rate of the
current is 5 mph.

3b. A piggy bank contains only nickels, dimes and quarters. The
value of the coins is $5.50. The number of nickels is three times
that of the dimes. The number of nickels is six more than twice the number
of quarters. Find the number of nickels, dimes, and quarters in the
piggy bank.

Answer:

Let n = the number of nickels
Let d = the number of dimes
Let q = the number of quarters

The value of the coins is $5.50:

The number of nickels is three times that of the dimes:

The number of nickels is six more than twice the number of quarters:

Putting this together in a system we get:

Simplifying the system we get:

Eliminating n from equations (1b) and
(2b) we get:

Eliminating n from equations (2b) and
(3b) we get:

Putting equations (4) and (5) together into a system of two equations
and two unknowns we get:

Eliminating d from equations (4) and
(5) we get:

Solving for q we get:

Plugging in 12 for q in equation (3)
and solving for n we get:

Plugging in 30 for n in equation (2)
we get:

There are 30 nickels, 10 dimes and 12 quarters.

Problems 4a - 4b: Solve each system by either the substitution
or elimination by addition method.

4a.

Answer:

I'm going to chose to use the elimination by addition method.
Note that this one would be very difficult to try to solve it by the substitution
method.

Eliminating the x squared terms we get:

Solving for y we get:

Plugging 2 in for y in the first equation
and solving for x we get:

Two solutions are (3, 2) and (-3, 2).

Plugging -2 in for y in the first equation
and solving for x we get:

Two more solutions are (3, -2) and (-3, -2).

The four solutions are (3, 2), (-3, 2), (3, -2), and (-3, -2).

4b.

Answer:

I'm going to chose to use the substitution method, however it would
be perfectly find for you to use the elimination by addition method.
Either way the answer will be the same.

Second equation solved for y:

Substituting the expression x for y into the first equation and solving for x:

This will not factor so we will need to use the quadratic formula
to solve:

Since, we have a negative in the square root, this means we did not
get real number answer for x.